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# The Essential Spectrum of the Essentially Isometric Operator

Published:2012-07-16
Printed: Mar 2014
• H. S. Mustafayev,
Yuzuncu Yıl University, Faculty of Science, Department of Mathematics, 65080, VAN-TURKEY
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## Abstract

Let $T$ be a contraction on a complex, separable, infinite dimensional Hilbert space and let $\sigma \left( T\right)$ (resp. $\sigma _{e}\left( T\right) )$ be its spectrum (resp. essential spectrum). We assume that $T$ is an essentially isometric operator, that is $I_{H}-T^{\ast }T$ is compact. We show that if $D\diagdown \sigma \left( T\right) \neq \emptyset ,$ then for every $f$ from the disc-algebra, \begin{equation*} \sigma _{e}\left( f\left( T\right) \right) =f\left( \sigma _{e}\left( T\right) \right) , \end{equation*} where $D$ is the open unit disc. In addition, if $T$ lies in the class $C_{0\cdot }\cup C_{\cdot 0},$ then \begin{equation*} \sigma _{e}\left( f\left( T\right) \right) =f\left( \sigma \left( T\right) \cap \Gamma \right) , \end{equation*} where $\Gamma$ is the unit circle. Some related problems are also discussed.
 Keywords: Hilbert space, contraction, essentially isometric operator, (essential) spectrum, functional calculus
 MSC Classifications: 47A10 - Spectrum, resolvent 47A53 - (Semi-) Fredholm operators; index theories [See also 58B15, 58J20] 47A60 - Functional calculus 47B07 - Operators defined by compactness properties

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